Non-Steady State Conduction

1. Non-Steady State Conduction

2. Non-steady State Conduction Goals: By the end of today’s lecture, you should be able to: • define the mechanisms for non-steady state conduction • determine the time required to transfer heat to and from: • flat plates • cylinders • spheres • describe the difference between constant surface combined convection and conduction.

3. Outline: • Conduction for a constant boundary surface temperature • Flat plate • Cylinder • Sphere • Conduction for a rate based boundary temperature • Flat plate • Cylinder • Sphere

4. Infinitely long solid slab (no end effects) (constant surface temperature) Heat Balance dx 2s Ts Ts Heat Flow Heat Flow

5. Divide by r cp A dx dt to yield: Where:a = thermal diffusivity = k/rcp Boundary Conditions:

6. Integrated Solution: Where: Ts = constant average temperature of surface Ta = initial temperature of slab Tb = average temperature of the slab at time t Fo = Fourier number = a tT/s2 a = thermal diffusivity = k/rcp tT time for heating s = one-half slab thickness a1 = (p/2)2

7. Neglect all but first term (for Fo greater than 0.1) and get:

8. For infinitely long (no end effects) cylinder: Where: Fo = a tT / rm2

9. For a sphere: Where: Fo = a tT / rm2

10. Constant surface temperature plot Figure 10.5 Average temperatures during unsteady-state heating or cooling of a large slab, and infinitely long cylinder, or a sphere.

11. A sphere – heat transfer at boundary function of convective rate Tf Ts

12. Biot number ( Bi) = convection / conduction Flat plate Cylinder and sphere

13. For a sphere at low Biot number: Assuming an effective internal coefficient and an overall heat-transfer coefficient

14. Conductive / convective mechanism plot Figure 10.7 Change with time of the average temperature of a slab with external convective resistance.

15. Conductive / convective mechanism plot Figure 10.8 Change with time of the average temperature of a sphere with external convective resistance.

16. Semi-infinite Solid x Ts Solid T at time t and position x

17. Semi-infinite Solid x Ts Solid Graphical solution to preceding equation T at time t

18. Problem Solution Matrix Geometry (sphere, slab, cylinder) Non-Steady State Problem Statement Steady State Calculate U,DT,Q,A Constant Surface (Ts) or Convective Film (Tf) Ts TAvg or T Position Tf T Position Calculate Uo or ho TAvg TAvg or T Position Resources Fig. 11.1.2 TAvg Fig. 10.5 Resources Fig. 11.1.3 T Position Fig. 10.7 Fig. 10.8 Eqn. 10.32 Resources Fig. (b-g)

19. Ten Minute Problem - The Thanksgiving Turducken I am cooking a 20 lb turducken (a turkey - stuffed with a duck - stuffed with a chicken – stuffed with stuffing – see photo below) for Thanksgiving dinner. How long will it take to cook ??? Initial temperature (T) of turducken on my kitchen counter = 70 F T oven = 350 F T of stuffing for a "done" turducken = 165 F External heat transfer coefficient for my Magic Chef natural circulating oven = 0.40 BTU / hr ft2 F Assume the turducken is a fat thing that approximates a spherical geometry. Volume = 4/3 p r3 Surface area = 4 p r2 Effective density of turducken = 65 lb/ ft3 Effective heat capacity of turducken = 0.83 BTU / lb F Thermal conductivity of turducken = 0.35 BTU / ft hr F